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In Mathematica the functions like Thread, Inner, Outer etc. are very important and are used frequently.

For the function Thread:

Thread Usage1:

Thread[f[{a, b, c}]]
{f[a], f[b], f[c]}

Thread Usage2:

Thread[f[{a, b, c}, x]]
{f[a, x], f[b, x], f[c, x]}

Thread Usage3:

Thread[f[{a, b, c}, {x, y, z}]]
{f[a, x], f[b, y], f[c, z]}

And I understand the Usage1, Usage2, Usage3 easily as well as I use them masterly.

However I always cannot master the usage of Inner and Outer so that I must refer to the Mathematica Documentation every time when I feel I need using them.

I find that I cannot master them owing to that I cannot understand the results of Inner and Outer clearly. Namely, I always forget what construct they generate when executed.

The typical usage cases of Inner and Outer shown as below:

Inner Usage:

Inner[f, {a, b}, {x, y}, g]
g[f[a, x], f[b, y]]
Inner[f, {{a, b}, {c, d}}, {x, y}, g]
{g[f[a, x], f[b, y]], g[f[c, x], f[d, y]]}
Inner[f, {{a, b}, {c, d}}, {{x, y}, {u, v}}, g]
{{g[f[a, x], f[b, u]], g[f[a, y], f[b, v]]}, 
 {g[f[c, x], f[d, u]], g[f[c, y], f[d, v]]}}

Outer Usage:

Outer[f, {a, b}, {x, y, z}]
{{f[a, x], f[a, y], f[a, z]}, {f[b, x], f[b, y], f[b, z]}}
Outer[f, {{1, 2}, {3, 4}}, {{a, b}, {c, d}}]
{{{{f[1, a], f[1, b]}, {f[1, c], f[1, d]}}, 
  {{f[2, a], f[2, b]}, {f[2, c], f[2, d]}}}, 
 {{{f[3, a], f[3, b]}, {f[3, c], f[3, d]}}, 
  {{f[4, a], f[4, b]}, {f[4, c], f[4, d]}}}}


  1. How to master the usage Inner and Outer? Namely, how can I use them without referring to the Mathematica Documentation?

  2. How to understand the result of Out[3],Out[4],Out[5] figuratively? Namely, by using graphics or other way.

share|improve this question
I recommend that you download and work through Leonid Shifrin's Mathematica programming: an advanced introduction. It's free and answers a lot of question you ask. – m_goldberg Aug 20 '14 at 10:12
@m_goldberg,I read the 1-3 chapters of that book three months ago,but owing to some other things, I don't read the latter chapters.Thanks for your suggestion sincerely.I will continue to read it right now.:-) – Shutao TANG Aug 20 '14 at 10:23
up vote 18 down vote accepted

I think of Outer just like nikie showed.

Inner is a generalization of matrix multiplication. I like the picture from the Wikipedia page.

Matrix Multiplication

To calculate an entry of matrix multiplication, you first pair list entries (a11,b12) and (a12,b22). You "times/multiply" those pairs (a11*b12) and (a12*b22), and then you "plus/add" all the results (a11*b12)+(a12*b22). Note that you "times" before you "plus" in matrix multiplication which helps me remember the order of arguments for Inner.

share|improve this answer

Animated Mathematica Functions contains cool animated illustrations of the way a number of built-in functions work. Among them are


enter image description here


enter image description here


enter image description here

See also: cormullion's video

share|improve this answer
@kguler...I am learning so much this week...nice – ubpdqn Aug 20 '14 at 12:52
thank you @ubpdqn... me too :) – kglr Aug 20 '14 at 13:08
Loved them when they first came out; still love them today. – J. M. May 4 '15 at 10:02
@Guesswhoitis., same here -- especially the sound effects:) – kglr May 4 '15 at 10:10
@Guesswhoitis., I know you are J.M :) Welcome back! – Shutao TANG May 5 '15 at 5:24

Not sure if that's what you're looking for: This is the image I always have in mind for Outer[f,{a,b,c},{x,y,z}]:

enter image description here

args = {{a, b, c}, {x, y, z}};
TableForm[Outer[f, args[[1]], args[[2]]], TableHeadings -> args]
share|improve this answer

I think of Outer like nikie's answer shows. Here's a similar view of Inner. Think of the arguments in columns. Apply f to each row and g to the result.

Mathematica graphics

args = {{a, b, c}, {x, y, z}};
Format[g[e__]] := Column[{g, e},
   Dividers -> {None, {False, True, False}}, Alignment -> Center];
Inner[f, Sequence @@ args, g]
share|improve this answer
Might I suggest f@@{a,x} etc.? – Timothy Wofford Aug 20 '14 at 13:49
Thanks. I wanted a divider, but I hate dealing with tables/grids in Mma. I'd thought about f[a, x], too (i.e., no Format-ting). I was trying to emphasize the columns. – Michael E2 Aug 20 '14 at 13:52
(i = Inner[List, Range@3, Range@3, List]) // MatrixForm;

enter image description here

(o = Outer[List, Range@3, Range@3]) // MatrixForm

enter image description here

p1 = ListLinePlot[i, Mesh -> All, PlotStyle -> Red, PlotTheme -> "Detailed"];
p2 = ListLinePlot[o, Mesh -> All, PlotStyle -> Blue, PlotTheme -> "Detailed"];

Legended[Show[p2, p1, PlotRange -> All], LineLegend[{Red, Blue}, {"Inner", "Outer"}]]

enter image description here

share|improve this answer
like you answer+1 – ubpdqn Aug 20 '14 at 12:04

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